teaching, math, teaching math

Discovery

I don’t believe that students necessarily learn better when they figure things out for themselves.

To clarify, I’m not trying to argue that kids shouldn’t have a chance to figure things out, or that it’s inherently bad. Just that, in and of itself, student discovery is not a particular priority for me. Some reasons:

Cognitive Load TheoryCognitive Load Theory is fascinating to me. It’s also interpreted lots of different ways. Fundamentally, it says that while trying to figure something out, working memory resources are consumed in such a way that it is difficult to move information to long term memory. I’ve definitely seen this happen. Kids spend a great deal of effort searching for a possible solution to a problem and meet various dead ends along the way. By the time they reach a viable strategy, much of their thinking has been spent at cross purposes with the goal of the lesson. That realization they make at the end is just one synapse firing over a long lesson and isn’t particularly durable.

I’m skeptical this is the case in every instance, and I think there’s a lot more subtlety to things than “increased cognitive load of discovery = bad”. But it’s something important to think about — in the words of Ben Blum-Smith, “any thoughtful teacher with any experience has seen students get overwhelmed by the demands of a problem and lose the forest for the trees”.

How much does one activity matter?
I tried hard to create powerful discovery experiences early in my career. An implicit belief embedded in that instruction was the idea that, if I could just find the perfect way to introduce students to an idea, they would remember it and be able to apply it in the future. At best I had mixed success. One activity, no matter how clever it is, never makes as much of a difference as I might think. The sum of student experiences — from the introduction of an idea, through practice activities, to opportunities to transfer that understanding to new contexts — are what make a difference for learning. I’m much more interested in focusing my energy on a range of activities that allow students to practice and extend the ideas we’re learning about than putting all my eggs in the “they discovered it so they’ll remember it” basket. Here’s my core value: it matters less how a student figures our something new than what they do with that knowledge in the future.

I do still use discovery-oriented activities. Here are two goals I have that this type of lesson can effectively support.

Intellectual NeedOne idea that pushes back against an aspect of Cognitive Load Theory is the generation effect. In short, trying to figure something out before learning it leads to more durable learning. I want learning to be active for students whenever possible. I want to avoid overwhelming them, but I prefer active tasks whenever possible, and if the task is within students’ reach, I’m likely to use it. In addition, I want to create some intellectual need for what students are going to learn. If struggling with a problem makes students aware of what they don’t know and what that knowledge might be useful for, they are well-positioned for future learning.

Neither of these strategies requires that students actually reach the big ideas of a lesson on their own or struggle for a great deal of time; they just argue that attempting to do so to begin a lesson can lead to more durable learning. And, in both cases, the sequence ends with an opportunity to make the learning explicit and consolidate understanding. That explicit instruction doesn’t have to come from me. One way I often do this is to use group work to provide opportunities for students to share strategies, and lead into a full-class discussion sharing those strategies and consolidating the big ideas that students need to move forward. That discussion can be largely student driven. But the consolidation is essential; it’s incredibly rare for me to see every student figure something out in a discovery lesson. And, honoring the principles of Cognitive Load Theory, even if many students figure something out on their own, further clarification and exposure beyond a discovery activity is essential to further cement their understanding.

Wonder
I love doing math because of the joy of learning something new through discovery. This doesn’t mean that students necessarily feel the same way or learn best that way. But it does mean I want students to have the opportunity to experience the wonder and joy of mathematics. Even if creating these experiences is an inefficient use of class time or leads to less learning, it’s a priority for me. Not every day and often not every week, either. Being judicious with how often I use these activities allows me to focus on the ones I do and do them right. And watching a student light up when they have that moment of insight is a special thing to see and a special thing to experience.

Conditions for Discovery

A discovery activity should either focus on an incremental, manageable step forward or something so mathematically spectacular that it is worth significant effort

I must be willing to cut an activity short if it’s clear we’re hitting a dead end

An activity has to end with time spent consolidating understanding, either student driven or teacher driven

I won’t choose a discovery activity at the expense of time spent on practice activities that allow students to deepen their understanding, and allow me to see who understands and who doesn’t and adjust future instruction appropriately

I don’t mean to get too down on discovery. Instead, I want to clarify more specific and concrete goals than “students figure things out themselves”. For me, the goals of intellectual need and wonder are worth working toward, and are much more connected to experiences and outcomes I care about than discovery for its own sake.

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21 thoughts on “Discovery”

I enjoyed reading this, possibly because it agrees with many of my opinions and observations. 😉 Pure discovery learning has been shown to be very inefficient use of student time, for the reasons you gave and the fact that sometimes they discover the wrong things. However guided discovery (contrasted with leaving students to do it on their own) can be a great way to learn, as roadblocks are avoided, and the cognitive load you describe is reduced. I take on board the need to pull it all together at the end also. I remind myself that learning involves understanding and remembering, and both these have to be addressed.

I pretty much agree with everything you say. The main and essential first step is to move away from all-discovery and no-discovery, as neither of those works consistently. Once you’ve done that, the work starts: for a given topic, how guided should the discovery be? what is needed to precede it? to complement it? to follow it? And so on. “Institutionalization” as they say in France is essential. That is the step where you *explicitly* bring students into the institution of mathematics: emphasizing the important ideas, introducing vocabulary and notation, etc. There are many ways to do this right, and there are many ways to do it wrong. The only way to get better at it is to pay attention to how specific activities play out in the real world.

But I’m not telling you anything you don’t already know, so I’ll shut up now. 🙂

I’m wondering if we haven’t been barking up the wrong tree for a long time with the term “discovery** learning.” That has often been misused as synonymous with “constructivist learning” and then one or the other of those as the umbrella term for a host of related but not identical ideas. In particular, I’ve seen people who are entirely or primarily fans of teacher-centered, direct instruction in mathematics try to suggest that what people who promote anything else are trying to do is to get every student to “rediscover” everything in the history of mathematics. And since that would take ludicrous amounts of class time, it can’t work.

I don’t think that’s our goal. I think we want students to have the OPPORTUNITY to think about many things in mathematics (not necessarily EVERY SINGLE THING) before they are told. Kamii and her colleagues stress this particularly in primary grades: the propensity has been in K-2 to just hand students “the” algorithm for adding, subtracting, etc., and heaven help those kids who don’t grasp it (and more importantly from that perspective, manage to do it quickly and correctly in short order). Apparently, there’s just no time for young children to come up with their own ideas, to test those ideas, to listen to the ideas of peers, to question the ideas of others, etc., before the teacher steps in to lay upon them the official method.

I believe that it’s inevitable that students who are solely or almost always taught math that way will be passive, will not trust their own ideas and intuitions, and will eventually stop trying to do anything in math that they have not been explicitly taught. And we have the evidence for that in about grade 3 on up. I don’t think the problem is cognitive load, or expecting kids to reinvent the wheel in mathematics, or anything like that. I think it’s learned helplessness and learned passivity in the primary grades that make kids disinclined to approach math from a “discovery” or “inquiry” approach. If we did better in the early grades, I believe we’d have less difficulty with students balking at opportunities for inquiry.

**I think “inquiry learning” is a better term. It takes away the notion that we’re trying to make students reinvent the wheel. And it also opens up the process to be less driven to a definite goal. If you are supposed to “discover” something, there has to be something that was “covered” you are supposed to find. If you’re “inquiring,” then everything is open. As for “wrong” discoveries, as mentioned by Dr. Nic, I’m not quite sure what that means. If it’s coming up with something that doesn’t hold up to further inquiry, that’s fine. Students think they have an answer that is wrong or a method that is wrong all the time (or would if we let them) and I see those as good opportunities for further inquiry and learning (and if we don’t want to pursue something for a good pedagogical reason, there are many good alternatives to handling errors and “wrong paths”). If it means that they found something that wasn’t the point, well, we have ways to deal with that, too. Sometimes, students come up with something unexpected but rich and very worthy of pursuit. Unless, of course, we’re being driven by harsh taskmasters who don’t want time spent on anything that isn’t going to be tested.

The language here is interesting. I deliberately avoided “constructivism” because I find folks who use that word to be more focused on processes than goals, which is what I’m interested in here. Similarly, I don’t see a necessary distinction between explicit instruction and active learning. Both can happen, at different stages in the learning process.

I agree with your point about what might be called “learned helplessness”; I also like inquiry as a value, but I find inquiry in and of itself to be a bit of a hollow goal for many of my students. Just my opinion.

dkane47 wrote, “I find inquiry in and of itself to be a bit of a hollow goal for many of my students. Just my opinion.”

Yes, and that is a part of my point about what happens to kids in the first few years of school when it comes to mathematics. They’re made passive and taught that their ideas and intuitions don’t matter: only the official knowledge passed down from the teacher (or textbook, or some other authority) matters.

But another part of what they are taught is that learning for learning’s sake has no value. Grades (or praise, or other rewards – material or symbolic) are the point of school. Learning is almost the booby prize. So inquiry? Just thinking, exploring, “playing” with knowledge and questions? All part of a fool’s errand. Why do more than the minimum necessary to get the grade you’ll be satisfied with (which varies from student to student, of course)?

Not every single kid falls prey to the above, but I believe it gets to most. It got to me early on. It took a very long time to work past that. I can’t swear that even now it doesn’t still taint me.

I don’t think that’s natural. I don’t think it reflects our in-born need to make sense of our world. I think it’s part of what the institutionalization (that word again) of learning does to human inquiry and turns it into quid pro quo: I’ll do this if you give me that. Teachers who want to be more inquiry-oriented in higher grades get frustrated by all the difficulties they face getting “alternative” instructional methods to work for some or many kids. But that isn’t because the ideas about human learning are wrong. It’s because we warp kids so much that by the time they leave elementary school, few still hunger to learn because it’s gratifying to understand more about the world. They either are playing a game to get by at some level or they are trying to beat everyone else from a competitive standpoint. That, too, makes cooperative and inquiry learning difficult.

I love that word, “institutionalization”, similar to how I’ve come to like “explicit instruction” more and more. Though I do want to reiterate that it doesn’t necessarily have to come from me, depending on how much of the class has reached the big ideas.

A lot of this has underlying unanswered questions about what’s going on emotionally in that “discovery” phase. It’s pretty nigh onto impossible to build an “intellectual need” if a student is still worried about… whatever… and it’s a real thing that somehow gets glossed over in theory. Your attention to cognitive load and adding structure to things goes a long way to help with that.

Agreed. Which can lead to some troubling assumptions about certain students and whether they can or can’t do math. Being willing to try in these contexts may be most of the battle, and there are a ton of factors going into that.

I tend toward more discovery-oriented activities than some do, but I do agree that an overly discovery-centric approach CAN make it harder to learn the skills of a math classroom. I know that students don’t end my heavily guided-discovery-centric geometry course any more efficient at, say, solving right triangles than they would in a more structured teach-practice environment.

However, my personal frustrations with math education, and the reason I keep using more discovery-centric work, is that I really want students to more thoroughly understand WHY certain things are true – sine, cosine, tangent are just funny names for similarity ratios between right triangles and we could measure them ourselves and use our measured values forever – and my experience is also that students somewhat better retain that reasoning when they have to build it up themselves, rather than having it thrust upon them. A proof is more powerful if you do it yourself, even if guided through it, than if you are shown it.

I’m particularly interested here in the activities teachers use to introduce a new concept. I wonder whether the choice of an activity for that specific part of the learning process is most important for understanding, or if we can do that effectively bit by bit, over time.

As an example, I was frustrated introducing the unit circle this year when we did an activity that had students generate the various parts of the unit circle using right isosceles and equilateral triangles. Just three days later, many students were unable even to reproduce a proof of where certain values of sine and cosine come from. I introduced that topic in a largely discovery manner, but it didn’t seem to particularly support understanding, and I think I needed to reinforce those ideas throughout the unit and not just hit on them once.

I think you make a good point about understanding, but my experience is that it’s more complicated than a choice of activity to introduce a new idea.

Like Dr Nic mentioned, pulling it all together at the ends is the essential difference I’ve found between widespread understanding and widespread misconceptions. Building the need for new math tools shouldn’t take long, and sometimes discovering the technique doesn’t take too long either. Going down a few dead ends can be productive as well. I’ve had a lot of success with discovery activities, even ones that seemed to fail or go to unexpected places, as long as a good and meaningful reflection takes place afterwards. A written component also helps after a class discussion.

Briefly, I’ve found a combination of discussion and personal written reflection the same day, then reviewed 1 day, 1 week, and ideally 1 month and more later. Discussion helps all students come to agreement (sometimes with guidance) on the key concepts or strategies developed. Personal reflection helps them take ownership and makes them accountable.

I think that my only addition or caveat to your post, Dylan, is to push back a bit on the goals of math class. If the primary or only goal is remembering/applying mathematical content knowledge, then your post makes complete and total sense – we should probably use discovery sparingly; it is helpful as a motivator (basically, the intellectual need and wonder categories you listed) and maybe helps some students remember some ideas some of the time. But, if one’s goal is to teach students to think like mathematicians, then I don’t know of a better way than having them engage in the process of doing math consistently and frequently while also seeing models of what this might look like and getting feedback on their efforts and ideas. I don’t think that anyone would argue that a discovery approach is the most efficient method of transferring knowledge, but for me at least, that’s not the primary goal.

I hear that, and I could have made that explicit when I talked about wonder. I think that both the feeling of doing math and the skills of mathematicians are important goals. But here’s my hesitation, and maybe this is particular to my experience:

My experience is that, when I ask students to do something ambitious mathematically, it is often unsuccessful. In particular, too many of those lessons are successful for a subset of students, but unsuccessful for others and it tends to be the same kids. Those kids feel unsuccessful because they couldn’t figure it out, learn much of what they learn from peers in group work, or become frustrated from the activity, don’t listen to the discussion at the end, and fall behind.

I’ve shifted to an approach that relies less on discovery of new ideas, and more on ambitious application of those ideas. If I want students to look for and make use of structure, or model with mathematics, or look for and express regularity in repeated reasoning, I don’t see a particular reason why that has to happen when students are learning new content. And I see more students able to be successful when I can do some work to “level the playing field” before we try to exercise these habits of mind.

This is a bit more push-back-y than it needs to be. I definitely agree with what you’re saying. But I see multiple ways to get to those goals, and I’d like to avoid putting all my eggs in one basket.

Sharing this with my student teachers. I know this kind of patient thinking things through is common for you, but sometimes it really catches me by surprise anyway. I love how you get to the purpose of how you choose your lesson approach, and how there are so many purposes a math teacher might choose. Matching the purpose, the content and the students is always going to be a challenge. But I love that challenge!

Thanks, John. I think two thirds of my improvement as a teacher has been asking myself “what are my goals today?” and “does that thing I thought of doing actually meet those goals?”. Important, hard thinking to do.

It’s interesting to see the different ways we dichotomize this issue in order to gain traction in our arguments. There’s “direct vs. discovery,” “instrumental vs. relational” (used by Skemp), “traditional vs. reform,” etc., and each has a different origin but nuances have sometimes been lost as they’ve been jumbled together. I just saw Anna Sfard use the terms “ritualized vs. explorative,” which is a new distinction to me. Rather than “discovery,” Freudenthal used the term “reinvention,” and with thinking similar to this post, he later adopted the term “guided reinvention” to stress the teachers’ role in knowing how, when, and why the reinvention should happen.

In recent years there have been multiple meta-analyses in science education that have helped bring some data to this discussion. One by Furtak, Seidel, Iverson, and Briggs (2012) found a positive effect of inquiry-based teaching reforms and the importance of teachers’ active guidance. Critics of inquiry-based teaching often equate it with discovery learning, although the latter has been mostly discredited in science education by those who see the distinction between the approaches. In the meta-analysis, the mean effect sizes for teacher-guided inquiry vs. traditional were more than twice as large as the mean effect sizes for “discovery”/student-led learning vs. traditional.